8. A two-person zero-sum game is represented by the pay-off matrix for player A shown below.
\section*{Player B}
Player A
| \cline { 2 - 4 }
\multicolumn{1}{c|}{} | Option X | Option Y | Option Z |
| Option Q | - 3 | 2 | 5 |
| Option R | 2 | - 1 | 0 |
| Option S | 4 | - 2 | - 1 |
| Option T | - 4 | 0 | 2 |
- Verify that there is no stable solution to this game.
- Explain why player A should never play option T. You must make your reasoning clear.
Player A intends to make a random choice between options \(\mathrm { Q } , \mathrm { R }\) and S , choosing option \(Q\) with probability \(p _ { 1 }\), option \(R\) with probability \(p _ { 2 }\) and option \(S\) with probability \(p _ { 3 }\) Player A wants to calculate the optimal values of \(p _ { 1 } , p _ { 2 }\) and \(p _ { 3 }\) using the Simplex algorithm.
- Formulate the game as a linear programming problem for player A. You should write the constraints as equations.
- Write down an initial Simplex tableau for this linear programming problem, making your variables clear.
The linear programming problem is solved using the Simplex algorithm. The optimal value of \(p _ { 1 }\) is \(\frac { 6 } { 11 }\) and the optimal value of \(p _ { 2 }\) is 0
- Find the best strategy for player B, defining any variables you use.